ODD + ODD = ODDIS IT POSSIBLE?llya Sinitsky, Rina Zazkis, and Roza Leikin explore odd andeven functions.

This article was inspired by the following observa-tion by a student:

When we started to deal vyith trigonometricfiinctions, the teacher mentioned that some ofthem are odd, or even. For example, the

ßinctiony = sin x is an odd fanction andy = cos X is an even function. But what is thereason for using the same adjectives 'odd' and'even' both for numbers and for functions? Itis simply confusing! These are really oddnames for functions!We decided to explore the reasons why some

functions are referred to as odd or even, and therelationships between these functions and odd andeven numbers. We start by recalling the definitions— see, for example, Larson & Hostetler, 2006:

Function^(x) of real argument x is even if, andonly if, the domain of the function is symmetricalaround 0 and the equalityJiJ—x) =f(x) holds forevery x from the domain.Functiony^(x) of real argument x is odd if, andonly if, the domain of the function is symmetricalaround 0 and the equalit)^^—x) = —J(x) holdsfor every x from the domain.

Simply stated, the value of an even function is thesame for a number and its opposite, whereas thevalue of an odd function changes for the oppositenumber when the argument is replaced by itsopposite. Of course, this requires that for eachnumber x its opposite is also in the domain of thefunction. According to the definitions, functions ofboth types are defined at symmetrical parts of thenumber line — or, for the whole domain of realnumbers, —^<x <°c.

Graphs of both odd, and even, functions aresymmetrical. The graphs of odd functions have arotational symmetry of 180 degrees with the centreat the origin of axes — also referred to as central

symmetry at the point of origin — since point (—a,—h) is on the graph if, and only if, (a, b) point ison the graph of the function. Thej'-axis is thesymmetry line of the graph of every even function,since point {~a, b) is on the graph if, and only if,point (a, b) is on the graph of the function.

However, what reason is there for giving thename 'even' to the functionj = cos x (Figure I), orfor the function J = '1 —x (Figure 2)1 Perhaps thisis due to seeing the graph of an even function as aunion of two congruent — up to mirror reflection —parts, as analogous to seeing each even number asthe sum of two identical natural numbers?

-6 \ /V/: :-1.5 -

\ /

Figure I: y = cos x is an evenßjuction

Figure 2:y = '1 — x' is an evenßinction

10 MATHEMATICS TEACHING 225 / NOVEMBER 2011

However, the graphs of odd functions may alsobe considered as a union of two congruent partswhose values are opposite for opposite arguments.For example, function j = sin x is an odd function(.see Figure 3). This view of the odd function doesnot provide an analogy with odd numbers. Thus,we turn to well-known odd and even functions inwhich relationships between the functions and thenumbers can be clearly seen.

Figure 3: y = sin x is an odd function

Functionsy(x) = x" for natural values of n.It is evident that functionj^x) = x" for every

even natural number (n = 2m) is even, since itholds the equality:/(-x) =/(x).

/ ( -x) =(-x)" =(-x)2'" = ( - 1)2".. x2" = X" =/(x)

Among the examples of even functions, one caneasily see the quadratic function, j = x , or thefunction y = x'^ whose graphs — parabolas ofdifferent degrees — are symmetrical with respect tothe^-axis.

For every odd natural number n = 2m + 1, thefunction^(x) = x" is an odd function: (—x) = ~f(x).

-X" = -/(x)

In particular, the simplest linear functionj = x— where the exponent of the variable is equal to 1— and^ = x' are odd functions whose graphs are.symmetrical with respect to the origin of thecoordinate system.

Since the notion of even and odd numbers canbe expanded from the natural numbers to integers,this leads us to consider integer exponents. Forn = Q, the functionjl x) = x" becomes constant:

f(x) = x" = 1, and it is an even function in thesense of symmetry with respect to the j-axis.

In the next step we focus on functionsf{x) = x~" with natural values ofn, and take into

account the well-known definition: x—" = ^n. Thebehaviour of these functions with negative integervalues of exponents differs significantly from thepolynomial functions with natural exponents. Forexample, they are not bounded in the neighbour-hood of the point x=0, and are not defined at thispoint. Nevertheless, the symmetry of thesefunctions fits their names: functionsj((x) = x -"have aj-axis symmetry for even values of n, andcentral symmetry for odd exponents, even thoughthe centre of symmetry (0;0) does not belong tothe graph of the function — .see Figure 4. The la.stcase also reminds us that the origin of the coordi-nate system does not necessarily belong to thegraph of every odd function.

-1 n 1

(A)

1

i11

1i

i

1

An 1 ^-1 ^

(B)

Figure 4: Graphs offunctions y = x" and y = j^nfor: (A) even valúes of {n > 0); (B) odd values ofn( n> l )

As can be derived from these examples, the useot the tags 'even' and 'odd' for functions comes asan extension of well-justified terms for even or oddexponents. Since all the functions of the form

^(x) = x" with even n have symmetry with respectto j-axis, it is a logical extension to name all thefunctions with this symmetry as 'even' ones — and asimilar analogy can explain the use of the term'odd' for functions with central symmetry

Furthermore, each linear combination of evenexponents of a variable, such as j = x'* — 3x2 -|- 5(Figure 5) or y = 3.2x12 + ¡j jio -|- x^, provides anadditional example of an even lunction — and the

Figure 5: y = x'' — 3x'? + 5 is an even function

MATHEMATICS TEACHING 225 / NOVEMBER 2011 31

-3 i -1V> (

• -6 -Figure 6: y = x^ — Sx^ + 4x is an odd function.

10 ;

I n

-I -1 C

-5 -

k f

1 1 2

figure 7:y = x^ — 3x^ + S is neither an odd nor anevenfunction.

analogy is complete for any linear combinations ofodd exponents, such asj = x — Sx^ + 4x (Figure6) or y = 3.2x1' + Wx^ - x^ , but not fory = x^ — ix^ + S (Figure 7), which can be apossible assessment task for students.

While extending the terminology based onsymmetry appears a plausible, logical explanationfor the choice of even and odd descriptors forfunctions, additional reasons can be found byconsidering more advanced mathematics. We turnto those in the next section.

Considering Maclaurin seriesAn arbitrary infinitely differentiable function can berepresented with Maclaurin — power — series. Fordetails see for example, www.intmath.corn/Series-expansion/2 Maclaurin-series.php.

The Maclaurin series is a sum ot non-negativeinteger powers of a variable with coefficientswhich are determined by the dérivâtes of thefunction at the point x = 0:

f(x)

Generally, this series approximates the functionin some interval near the point ;£=0.• What does this series look like for an even

function?• What is the value of its derivativeJ '(x) forx=0?

According to the definition.

but if this value does exist, for an even function it isnecessarily equal to its opposite value:

/'(O) ^ lim.-o- A x - A x

AxIt means that^'(O) = 0. Moreover, a very similar

argument leads to the equalityy(x) = ~f'(x) forevery point of differentiability of the initial function.Thus, the derivative of an even function is an oddfunction.• What about the derivative of any odd function:

= A

we can deriveû'(-x) = lim

Ai-O

- hmAx-O

Ax

Ax

—Ax

2! 3!

\ \ u , ^ AAx-l) — A x

— because the last ratio is just the definition ofg'(x), since ( —Ax)-*O is equivalent to Ax -* 0.Thus, the derivative of an arbitrary differentiableodd function is an even one. Now we turn back toan even function:^(x).

The first derivative of an even function is anodd function. What about its second derivative?The second derivative is a derivative of the firstderivative function, but the functionJ '(x) is an oddfunction, hence its derivative, i.e. (f'(x))' =f"(x) isan even function. This chain of arguments can bedrawn further; thus derivatives of odd order (f'(x),

f"'(x),f''''i(x), ... of differentiable even functions areodd functions, and derivatives of each even order(f"(x),f("''>(x), ...) for differentiable even functionsare even functions. It means that forx = 0

/'(O) =/"'(0) =/C)(0) = ... =/(2"+i)(0) = 0In other words, all odd exponents of Maclaurin

decomposition of an even function have zerocoefficients and, in fact, this series consists of evenpowers of the variable only Correspondingly, thesecond, the fourth, the sixth, and every even-orderderivatives for odd functions are odd functions with

MATHEMATICS TEACHING 225 / NOVEMBER 2011

/"(O) =/(">(x) = ... =/(2")(x) = OTherefore, the Maclaurin decomposition of oddtunctions consists of odd powers of the variable only.

Beyond calculusHistorically, continuous and differentiable functionshave been treated within Calculus, and — as shownabove — the reason for using the terms 'eventunction' and 'odd function' are far beyond theformal extension of names of polynomial functions.Nevertheless, we apply those names also to alltunctions that fit the formal abovementioneddefinition. In the family of even functions we cansee the familiar absolute value tunction J(x) = \x\that is not differentiable atx=O. In the family ofodd functions we notice the function —

- l , / o r x < 0ijor x = 0, which is non-continuous ati=O.

l,/orx>0sgn(x)= < 0,

I 1

Figure 8: y = sgn(x) is an odd Junction.

Moreover, the Dirichlet function 9(x) — definedas 6(x) = 1 for each rational x and 6(x) = 0 foreach irrational x — provides an example of an eventunction that cannot be plotted, but holds the j-axissymmetry — since the rationality or irrationality ofa number x is the same as that of its opposite (~x).

Considering arithmeticoperationsIs there an analogy between odd/even numbers andodd^even functions with respect to arithmeticoperations? If we start with addition and subtrac-tion of even tunctions, the analogy appears to hold:the sum, and difference, of even functions is aneven tunction with tull correspondence to theaddition or subtraction of even numbers. However,the sum and difference of odd functions is an oddtunction — contrary to the expectation by analogywith the set of integers. Extending the investigationto the multiplication and division of functions canbe an appro])riate task for students using, forexample, graphing technology; The results ofaddition, .subtraction, multiplication and division,where possible, ot functions are summarised in thefollowing table.

+,-

Evenfunction

Oddfunction

Evenfunction

Evenfunction

Oddfunction

Oddfunction

Evenfunction

Oddfunction

Evenfunction

Evenfunction

Oddfunction

Oddfunction

Oddfunction

Evenfunction

Figure 9: Outcomes oJ arithmetic operations with odd and even Junctions.

As seen in Figure 9, arithmetic operations witheven and odd tunctions are not in accord withoperations with even and odd numbers. However,the most surprising result is the pair of empty cells.We focus on this result in the next section.

Limits of analogyWhat is the meaning of the empty cells in theaddition/subtraction table? When an odd functionis added to an even one — both non-zero, what canbe said about the result? Surprisingly or not, theresult is neither odd nor even. And this is the maindifference between even and odd integers and evenand odd functions. Each integer is either even orodd, and the .set of integers is the union of twosubsets with an empty intersection. This is totallydifferent with functions. First of all, a functionexists which is both even and odd. Eor such afunction the following should hold for every x inthe domain:

J(-x) =/(x) and/(-x) = -/(x) means/(x) = -J(x)This is possible only (OTJ(X) = 0 for any

symmetric domain.More importantly, though, most functions are

neither odd nor even. Consider the simple exampleof the sum of the tunctionJ((x) = x and the constantfunction g(x) = 1. The graph of the sum of thesefunctions/ =/(x) + g(x) = x H- 1 is not symmetric,neither with respect to the j-axi, nor with respectto the origin of the axes.

Striving for symmetrySymmetry properties of even and odd functions arevery attractive. Is it possible to use an arbitrarytunction to construct an even or an odd function?The geometric interpretation of oddness andevenness can assist us in achieving this goal. Whentunctiony(x) is even, the equalityj^x) =J(—x) holdsfor every point x from the domain of the function.If the domain of the function is a symmetric one —it includes for each point, xo and its opposite, —xo.

MATHEMATICS TEACHING 225 / NOVEMBER 2011 35

this function can be modified to be even. The'non-evenness' of any function for each argument xis caused by the non-zero differenceA(x) =f{x) -f(-x).

To achieve equality A(x) = 0 we define thevalue of the new function at this point as an arith-metical mean of the values, Ej{x) = i (f{x) +f{—x))(see Eigure 10). In fact, it is easy to see that for eachfunctiony^(x), E^ (x) is an even function:Ej (-X) = i (f{-x) +f{-(-x))} = i (f(-x) +f(x))

1 1 T U I I ^1

- 3 - 2 - 1 0 1 2 3

Eigure 10: Constructing Ef(x) = i (f(x) +f(-x))forf(x)=\x+l\

It makes you think . . .George Knight pauses for thought twice ...1 This T-shirt legend was seen in Regent Street, London,

in the summer of 2011.

There are i o sorts ofpeople in the world.

Those who countin binciry

aryd those who don't.

2 Alfie - aged five years and two months - responds toa comment from his granny by saying: "That's not afew, that's five - three is a few!"

There might just be something here worth pondering forsay, just a few moments ...

By very similar arguments, for each functionwith a symmetric domain an odd function can beconstructed: Ojix) = Hf{x) -f{-x)).

For 'pure' even or odd function these newfunctions hold special properties:

When functionJ^(x) is even, Ef (x) = (f(x) andOj(x) - 0.When function^(A) is odd. Of (x) = (fix) and

On the other hand, an arbitrary iunction^(x)with a symmetric domain may be presented as asum of even and odd components of this function:Ejix) + Ojix) = l(f{x) +/(-x)) + i(f{x) -fi-x))

Some analogy after allWe have considered arithmetic operations on evenand odd tunctions. However, one important opera-tion on functions so far has not been mentioned.This operation is a composition of functions; thatis, an application ot one function to the result otanother: g°f(x) = g(f(x)) with an appropriatechoice of domain for function^. Exploring thecomposition of even and odd functions — which isan appropriate task for students with or withoutgraphing technology — leads to the conjecture thatthe composition of functions upholds the analogywith multiplication of numbers.

The composition is an odd function when allthe components of this composition are oddfunctions:

-gof(x).For example, bothj = sin'x andj = sin(x3)

are odd functions.The composition of those tunctions — two or

more — is an even function when at least one of thefunctions is an even one. Here we are happy toleave the proof to the reader. For instance, all ofthese functions are even:

y = sin2x = (sinx)2 ; y = (cosx — 1)';y= \xM;y= (x2 -|-1)5.In summary, when you were convinced that

odd + odd = even, you were obviously thinkingabout odd numbers. With odd functions it is,without doubt, a different story.

llya Sinitsky works at the Gordon AcademicCollege of Education, Haifa, Rina Zazkis works atthe Simon Fraser University, Canada and RozaLeikin works at the University of Haifa.

ReferenceLarson, R. & Hostetler, R.R (2006). Precalculus. 7«'' edition.Brooks Cole.

MATHEMATICS TEACHING 225 / NOVEMBER 2011